Standing Wave and Harmonics Calculator: Vibrating String, Open and Closed Pipes and Beat Frequency

Strings, organ pipes and beats.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose the string, an open pipe, a closed pipe or beats.

Enter the length and the tension and density, or the speed of sound.

The calculator gives the fundamental and the harmonics.

Formulas used

  • f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}
  • f_n = \frac{nv}{2L}
  • f_n = \frac{(2n - 1)v}{4L}
  • f_{beat} = |f_1 - f_2|

Worked example

With the default numbers in the calculator above:

  1. Wave speed on the string: 328.165 m/s
  2. Fundamental frequency (first harmonic): 252.435 Hz
  3. Wavelength of the fundamental: 1.3 m
  4. The first five allowed frequencies (harmonics): harmonic 1: 252.43 Hz, harmonic 2: 504.87 Hz, harmonic 3: 757.3 Hz, harmonic 4: 1009.7 Hz, harmonic 5: 1262.2 Hz

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: What do you want to find? A pipe open at both ends.

  1. Fundamental frequency (first harmonic): 263.846 Hz
  2. Wavelength of the fundamental: 1.3 m
  3. The first five allowed frequencies (harmonics): harmonic 1: 263.85 Hz, harmonic 2: 527.69 Hz, harmonic 3: 791.54 Hz, harmonic 4: 1055.4 Hz, harmonic 5: 1319.2 Hz

Common questions

What is the fundamental frequency?

The lowest frequency at which the string or the air column can vibrate as a standing wave.

Why does a closed pipe only have odd harmonics?

It has a node at the closed end and an antinode at the open end, so only odd multiples of a quarter wavelength fit.

What are beats?

The slow rise and fall in loudness when two notes of nearly the same frequency are played together. The beat frequency is the difference between them.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any physics text (standing waves and harmonics).

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